Hi Matthew! That's great you are checking it out! The Transcendental Argument for God (TAG) is a fascinating apologetic and I'm glad to hear you are checking it out! I first ran into it at an Answers in Genesis summer conference in KY a few yrs ago in a presentation Dr. Jason Lisle gave. Been studying it ever since!
TAG essentially boils down to this core argument: only the biblical Christian worldview can account for the preconditions of inteligibility, thus proving the existence of the Judeo-Christian God. Granted, TAG can go farther than that, but that sums up a big thrust of it.
Dr. Lisle presented it by breaking down what the preconditions of inteligibility are. There are many, but he focused on 3: morality, uniformity of nature, and laws of logic. Ethics are fairly easy to utilize. An unbeliever may state, "Forcing your child to attend church should be considered child abuse!" A presuppositionalist could potentially reply, "I don't believe it is Child Abuse. God tells us to raise up our child in the way he should go! But if your world view is right, and there is no God, why is child abuse wrong?" It is an internal critique of the opponents worldview. Are ethical charges (I.e. "child abuse") consistent within the given worldview? Can said world view account for ethics in a logical manner? This is a weak example but it shows you the line presuppositionalists think along.
The method's founder, Cornelius Van Til, stressed that the ultimate point of contact between a believer and a non believer is the fact that both are made in the image of God. Both know God exists in their heart of hearts but "suppresses the truth in unrighteousness" (Romans 1). I
There are many great resources available online that provide good introductary material as well. One of the apologetics strongest proponents, the late Dr. Greg Bahnsen, has excellent videos on the topic. YouTube his name and "Defending the Faith".